Algebraic & Geometric Topology

A rational splitting of a based mapping space

Katsuhiko Kuribayashi and Toshihiro Yamaguchi

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Let (X,Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form Xαek+1 and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map α:SkX is greater than the Whitehead length WL(Y) of Y, then (Xαek+1,Y) has the rational homotopy type of the product space (X,Y)×Ωk+1Y. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dimX, then the mapping space (X,Y) can be decomposed rationally as the product of iterated loop spaces.

Article information

Algebr. Geom. Topol., Volume 6, Number 1 (2006), 309-327.

Received: 19 July 2005
Revised: 14 February 2006
Accepted: 14 February 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P62: Rational homotopy theory
Secondary: 54C35: Function spaces [See also 46Exx, 58D15]

mapping space $d_1$–depth bracket length Whitehead length


Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro. A rational splitting of a based mapping space. Algebr. Geom. Topol. 6 (2006), no. 1, 309--327. doi:10.2140/agt.2006.6.309.

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